LECTURE Notes.pdf

LECTURE
Notes
OMIS
2010
Jessica
Gahtan
OMIS2010 Lecture Notes Jessica Gahtan
Table
of
Contents
Lecture
1-­‐
January
8,
2013:
Intro
to
Linear
Programming
.......................................................
3
Page 2 of 21 OMIS2010 Lecture Notes Jessica Gahtan
Lecture 1- January 8, 2013: Intro to Linear Programming
- We need to realize that in business we will make decisions given a constrained environment and we want
to make the best decision in light of limitations you have in a given situation (that is, you want to use your
limited resources to do as well as possible)
The Decision Process
- Organizations have access to limited resources (time, space, etc.) which impacts the way that managers
meet their objectives
- Linear programming (LP) is one way that managers can determine the best way to allocate their scarce
resources; it’s a systematic process to come up with the best course of action (it can also identify some
other opportunities)
LP Applications
- Flexible, general structure - not limited to specific industry or area within business
- Aviation, military, businesses, transportation, facility location problem
Examples:
a) Scheduling school buses to minimize total distance traveled(transportation)
b) Allocating police patrol units to high crime areas in order to minimize response times to 911 calls
c) Scheduling tellers at banks so that needs are met during each hour of the day while minimizing the total
cost of labor
Mathematical Program ming
- A mathematical programming problem is one that seeks to optimize an objective function subject to
constraints → want to optimize- achieve goal as much as possible
Types of mathematical programming problems:
Linear programming – focusing on these primarily – the structure of the problem seems restrictive BUT a lot of
situations can be represented – get extra insights – advantages – foundation for exploring integer situations – have
more power
Integer Programming
Non-linear Programming – when a problem isn’t linear
Requirements of an LP Problem
1. LP problems seek to maximize or minimize some quantity (usually profit or cost) expressed as an
objective function
2. The presence of restrictions, or constraints, limits the degree to which we can pursue the objectives –
i.e. in deciding how many units of each product in a firm’s product line to manufacture is restricted by
available labor and machinery
• Can’t have unlimited profit because of restrictions like time, rules (i.e. labor law) wh ich hold us back
from achieving more
3. A feasible solution satisfies all the problem’s constraints
• Where the possible choices for decision variable are the different combinations that can work
4. An optimal solution is a feasible solution that results in the largest possible objective function value
when maximizing (or smallest when minimizing)
5. A graphical solution method can be used to solve a linear program with two variables
6. If both the objective function and the constraints are linear, the problem is referred to as a linear
programming problem
7. Linear functions are functions in which each variable appears in a separate term raised to the first power
and is multiplied by a constant (which could be 0)
• None of the variables can be squared or multiplied by each other – can be added to subtracted from
one another
8. Linear constrains are linear functions are restricted to be ‘less than or equal to’, ‘equal to’, or ‘greater than
or equal to’
• This is important- gives us something to test
9. Problem formulation or modeling is the process of translating a verbal statement into a mathematical
problem
Page 3 of 21 OMIS2010 Lecture Notes Jessica Gahtan
• Going from the description to story/ math
• While the math part is mechanical – the hard part is going from story to math- this part takes insight
Linear Programming
- Formulate the LP (Decision variables; objective function and constraints)
- Solution Methods (Graphical; Corner points method; excel solver; [Simplex])
Guidelines for Model Formulation
1. Understand the problem thoroughly
2. Describe the objective
3. Describe each constraint
4. Define the decision variables
5. Write the objective in terms of the decision variables
6. Write the constraints in terms of the decision variables
Example: LP Formulation
The Stratton Company produces 2 basic types of plastic pipe. Three resources are crucia l to the
output of pipe: extrusion hours, packaging hours, and a special additive to the plastic raw material.
Below is the next week’s situation.
Each unit of type1 yields $34, and each unit of type2 yields $40
Product
Resource Type1 Type2 Resource Availability
Extrusion 4hr 6hr 48hr
Packaging 2hr 2hr 18hr
Additive Mix 2lb 1lb 16lb
Limit: Resource Availability
Step
1:
Define
the
Objective
Maximize total profit
Step
2:
Define
the
Decision
Variables
X 1 amount of type1 pipe to be produced and sold next week
X 2 amount of type2 pipe to be produced and sold next week
Step
3:
Write
the
mathematical
objective
function
Max: $34 ∗ 𝑥 ▯ $40 ∗ 𝑥 =▯𝑧
Step
4:
Formulate
the
Constraints
Extrusion: 4𝑥 ▯ 6𝑥 ≤▯ 48
Packaging: 2𝑥 + 2𝑥 ≤ 18
▯ ▯
Additive mix: 2𝑥▯+ 𝑥 ≤▯ 16
Inequalities:
1. Typically the constraining resources have upper or lower limits
For example: For the Stratton Company, the total extrusion time must not exceed the 48 hours of capacity
available, so we use the ≤ sign
2. Negative values for constraints 𝑥 ▯nd 𝑥 d▯ not make sense, so we add the non -negativity restrictions
in the model: 𝑥 ▯ ≥ 0 and 𝑥▯ ≥ 0 (non negativity restrictions)
3. Other problems might have constraining resources requiring ≥,=,𝑜𝑟 ≤ restrictions
LP in the Final Form
Max 𝑧 = $34 ∗ 𝑥 ▯ $40 ∗ 𝑥 , ▯ubject to:
Page 4 of 21 OMIS2010 Lecture Notes Jessica Gahtan
1. 4𝑥▯+ 6𝑥 ≤▯ 48 (Extrusion constraint)
2. 2𝑥▯+ 2𝑥 ≤▯ 18 (Packaging constraint)
3. 2𝑥 + 𝑥 ≤ 16 (Additive mix constraint)
▯ ▯
4. 𝑥▯,𝑥▯ ≥ 0 (non-negativity constraints)
^ Equations are on the left side only, values of the constraint are on the right side only
Graphical Solution
Most linear programming problems are solved with a computer. However, insight into the meaning of the
computer output, and linear programming concepts in general, ca n be gained by analyzing a simple two -
variable problem graphically .
Graphical Method of linear programming:
A type of graphical analysis that involves 5 steps:
1. Plotting the constraints
2. Identifying the feasible region
3. Plotting an objective function line
4. Finding a visual solution
5. Finding the algebraic solution
Step
1:
Plotting
the
constraint
equations – disregarding the inequality portion of the constraints (> or < ). Making
each constraint an equality (=) transforms it into the equation for a straight line.
Step
2:
Identifying
the
feasible
region
The feasible region is the area on the graph that contains the solutions that satisfy all the constraints
simultaneously; to find it, first locate the feasible points for each constraint and then the area that satisfies all
the constraints.
- The feasible region for a two -variable LP problem can be nonexistent, a single point , a line, a polygon, or
an unbounded area
- Any linear problem falls into one of 3 categories:
1. Is infeasible
2. Has a unique optimal solution or alternate optimal solutions
3. Has an objective function that can b e increased without bound
- A feasible region may be unbounded and yet there may be optimal solutions. This is common in
minimization problems and is possible in maximization problems
Graphical Solution
Generally, the following 3 rules identify the feasible points for a given constraint:
1. For the = constraint, only the points on the line are feasible constraints
2. For the ≤ constraint, only the points on the line, and the points below or to the left of the line are
feasible constraints
3. For the ≥ constraint, only the points on the line, and the points above or to the right of the line are
feasible constraints
Page 5 of 21 OMIS2010 Lecture Notes Jessica Gahtan
Graphical Analysis - The Feasible Region
a) b)
c) d)
e)
Page 6 of 21 OMIS2010 Lecture Notes Jessica Gahtan
Find the Optimal Solution
We will demonstrate two approaches to find the optimal solutions:
1) Iso- Profit Line/ Level curve
2) Corner Points
Iso-Profit Line/ Level curve- Solution Method
Step
1:
Choose
a
possible
range
for
the
objective
function
𝑧 = $34𝑥 + $40𝑥
▯ ▯
204 = $34𝑥 +▯$40𝑥 ▯
Step
2:
Solve
for
the
axis
intercepts
of
the
function
and
plot
the
line
- A series of dashed lines can be drawn parallel to this first line.
- Each would have its own Z value: Lines above the first line would
have higher Z values; Lines below it would have lower Z values.
- Our goal is to maximize profits, so the best solution is a point on
the iso-profit line farthest from the origin but still touching the
feasible region
Find Point C:
I: 41 + 6x2=48 (extrusion)
II: 21 +2x2= 18 (packaging)
2x1+4x 2 30
x1+2x 2 15
x1= 15 - 2x 2
I: 4 (15-2x 2+6x =248
60-8x 2 6x =28
60-2x 248
x1=15-2*6=3
Z= 34*3+40*6=$342
Graphical Solution - Corner Points Approach
Now we want to find the solution that optimizes the objective function
- Even though all the points in the feasible region represent possible solutions,
we can limit our search to the corner points
Corner point: A points that lies at the intersection of two (or possibly more)
constraint lines on the boundary of the feasible re gion
- No interior points in the feasible region need be considered because at least
one corner point is better than any interior point
The best approach is to plot the objective function on the gr